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Theorem imass1 5948
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
Assertion
Ref Expression
imass1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem imass1
StepHypRef Expression
1 ssres 5862 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 rnss 5792 . . 3 ((𝐴𝐶) ⊆ (𝐵𝐶) → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
31, 2syl 17 . 2 (𝐴𝐵 → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
4 df-ima 5548 . 2 (𝐴𝐶) = ran (𝐴𝐶)
5 df-ima 5548 . 2 (𝐵𝐶) = ran (𝐵𝐶)
63, 4, 53sstr4g 3932 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3853  ran crn 5536  cres 5537  cima 5538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3402  df-un 3858  df-in 3860  df-ss 3870  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-cnv 5543  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548
This theorem is referenced by:  vdwnnlem1  16443  dprdres  19281  imasnopn  22453  imasncld  22454  imasncls  22455  utoptop  22998  restutop  23001  ustuqtop3  23007  utopreg  23016  metustbl  23331  imadifxp  30526  esum2d  31643  eulerpartlemmf  31924  bj-imdirco  35014  brtrclfv2  40921  frege97d  40946  frege109d  40951  frege131d  40958  hess  40974  resimass  42361  setrecsss  45906
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